Question: Simplify the following expression: $n = \dfrac{3p^3 + 21p^2}{18p^3 - 30p^2}$ You can assume $p \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $3p^3 + 21p^2 = (3 \cdot p \cdot p \cdot p) + (3\cdot7 \cdot p \cdot p)$ The denominator can be factored: $18p^3 - 30p^2 = (2\cdot3\cdot3 \cdot p \cdot p \cdot p) - (2\cdot3\cdot5 \cdot p \cdot p)$ The greatest common factor of all the terms is $3p^2$ Factoring out $3p^2$ gives us: $n = \dfrac{(3p^2)(p + 7)}{(3p^2)(6p - 10)}$ Dividing both the numerator and denominator by $3p^2$ gives: $n = \dfrac{p + 7}{6p - 10}$